Characteristic evolution has many advantages over Cauchy
evolution. Its one disadvantage is the existence of either
caustics where neighboring characteristics focus or, a milder
version consisting of crossover between two distinct
characteristics. The vertex of a light cone is a highly symmetric
caustic which already strongly limits the time step for
characteristic evolution because of the CFL condition. It does
not seem possible for a single characteristic coordinate system
to cover the entire exterior region of a binary black hole
spacetime without developing more complicated caustics or
crossovers. This limits the waveform determined by a purely
characteristic evolution to the post merger period.

Cauchy-characteristic matching (CCM) is a way to avoid such
limitations by combining the strong points of characteristic and
Cauchy evolution into a global evolution [25]. One of the prime goals of computational relativity is the
simulation of the inspiral and merger of binary black holes.
Given the appropriate worldtube data for a binary system in its
interior, characteristic evolution can supply the exterior
spacetime and the radiated waveform. But determination of the
worldtube data for a binary requires an interior Cauchy
evolution. CCM is designed to solve such global problems. The
potential advantages of CCM over traditional boundary conditions
are:

accurate waveform and polarization properties at
infinity,

computational efficiency for radiation problems in terms of
both the grid domain and the computational algorithm,

elimination of an artificial outer boundary condition on
the Cauchy problem, which eliminates contamination from back
reflection and clarifies the global initial value problem,
and

a global picture of the spacetime exterior to the
horizon.

These advantages have been realized in model tests but CCM has
not yet been successful in either axisymmetric or fully
three-dimensional general relativity. This difficulty may
possibly arise from a pathology in the way boundary conditions
have traditionally been applied in the Arnowitt-Deser-Misner
(ADM) [10] formulation of the Einstein equations which, at present, is the
only formulation for which CCM has been attempted.

Instabilities or inaccuracies introduced at boundaries have
emerged as a major problem common to all ADM code development and
have led to pessimism that such codes might be inherently
unstable because of the lack of manifest hyperbolicity in the
underlying equations. In order to shed light on this issue,
B. Szilágyi [137,
138], as part of his thesis research, carried out a study of ADM
evolution-boundary algorithms in the simple environment of
linearized gravity, where nonlinear sources of physical or
numerical instability are absent and computing time is reduced by
a factor of five by use of a linearized code. The two main
results, for prescribed values of lapse and shift, were:

On analytic grounds, ADM boundary algorithms which supply
values for all components of the metric (or extrinsic
curvature) are inconsistent.

Using a consistent boundary algorithm, which only allows
free specification of the transverse-traceless components of
the metric with respect to the boundary, an unconstrained,
linearized ADM evolution can be carried out in a bounded domain
for thousands of crossing times with robust stability.

The criteria for robust stability is that the initial Cauchy
data and free boundary data be prescribed as random numbers. It
is the most severe test of stability yet carried out in the
Cauchy evolution of general relativity. Similar robust stability
tests were previously successfully carried out for the PITT
characteristic code.

CCM cannot work unless the Cauchy code, as well as the
characteristic code, has a robustly stable boundary. This is
necessarily so because interpolations continually introduce short
wavelength noise into the neighborhood of the boundary.
Robustness of the Cauchy boundary is a necessary (although not a
sufficient) condition for the successful implementation of CCM.
The robustly stable ADM evolution-boundary algorithm differs from
previous approaches and offers fresh hope for the success of CCM
in general relativity.